Algebraic Solution to Optimal Scheduling Problems Taking into Account the Scheduled Start Time of Jobs in Projects

A direct analytical solution is proposed for problems of the optimal scheduling of jobs within a project, which is based on the models and methods of tropical (idempotent) optimization. Optimal scheduling problems are reduced to the problems of tropical optimization, which consist in minimizing the objective function under given strict constraints on the start and finish times of jobs. The maximum deviation from the scheduled times for the start of the jobs within a project, which should be minimized, is taken as the optimal scheduling criterion. Strict constraints on the execution time of jobs within a project are given in the form of precedence relations and bounds on the start and finish times of jobs. Such problems arise when, for one reason or another (for example, due to technological limitations or safety requirements), it is necessary to start jobs at a specified time. In the work, constraints and objective functions are first described in terms of ordinary mathematics, and then project scheduling problems are formulated. Then elements of tropical mathematics are presented, which are necessary for optimal scheduling in a tropical form, and the scheduling problems are formulated in terms of idempotent mathematics and reduced to a tropical optimization problem. A solution to the problem is proposed in an explicit analytical form, which is well suited for both formal analysis and numerical computations. An illustrative numerical example is provided at the end of the work.